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双平方定理动态演示

双平方定理动态演示
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2017-04-13 10:31
向量三角函数解析几何变换几何作图
①本范例动态演示了著名的“双平方定理”，通过图形变换直观展示定理的几何意义；②涉及的核心数学概念包括平移、余弦定理、缩放、三维向量、算术平均数等；③作图要点提炼自构造摘要，使用了Polygon（多边形）、Segment（线段）、Dilate（缩放）、Rotate（旋转）、Circle（圆）、Ray（射线）、Intersect（交点）、Point（点）、OrthogonalLine（垂线）、Line（直线）、Midpoint（中点）、CircleArc（圆弧）、CircleSector（扇形）、Vector（向量）、Translate（平移）、Angle（角度）、Mirror（镜像）等GeoGebra命令与对象；④适合高中及以上数学学习者，用于理解几何变换与定理证明的动态过程。

GeoGebra 命令及参数

  1. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  2. Segment(B_2, C) → h
  3. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  4. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  5. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  6. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  7. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  8. Segment(R, A''') → f''
  9. Segment(A, B_2) → g
  10. Circle(B_2, g) → q_2
  11. Ray(A, B_2) → m_2
  12. Intersect(q_2, m_2, 2) → J_1
  13. Segment(B_2, J_1) → n_2
  14. Point(n_2) → K_1
  15. Segment(B_2, K_1) → r_2
  16. Circle(K_1, r_2) → t_2
  17. OrthogonalLine(K_1, r_2) → s_2
  18. Intersect(t_2, s_2, 2) → L_1
  19. OrthogonalLine(L_1, h) → f_3
  20. Intersect(f_3, h) → M_1
  21. Dilate(A, If[17 < e ≤ 18, 18 - e, 0], M_1) → A'_4
  22. Segment(A, A'_4) → j_3
  23. Line(K_1, M_1) → i_2
  24. Intersect(i_2, f'') → V_1
  25. OrthogonalLine(V_1, j_3) → n_3
  26. Intersect(n_3, j_3) → C_2
  27. Midpoint(A, C) → V
  28. CircleArc(V, C, D) → g_2
  29. Point(g_2) → B
  30. Segment(C, B) → j_1
  31. Segment(A, B) → n
  32. Intersect(n, f'') → Z
  33. Segment(B, Z) → l_1
  34. Segment(A, A') → f
  35. Segment(A'_1, A'') → f'
  36. Segment(C, D) → i
  37. Circle(B_2, (0.2 * r_2)) → p
  38. Segment(D, A) → j
  39. Circle(C, Radius[p]) → k
  40. Circle(A, (0.1 * j)) → q
  41. Circle(D, (0.1 * j)) → r
  42. Segment(A, C) → l
  43. Segment(D, B_2) → m
  44. Intersect(r, j, 1) → F
  45. Intersect(r, m, 1) → G
  46. Intersect(r, i, 1) → H
  47. Intersect(k, i, 1) → I
  48. Intersect(k, l, 1) → J
  49. Intersect(k, h, 1) → K
  50. Intersect(p, h, 1) → L
  51. Intersect(p, m, 1) → M
  52. Intersect(p, g, 1) → N
  53. Intersect(q, g, 1) → O
  54. Intersect(q, l, 1) → P
  55. Intersect(q, j, 1) → Q
  56. CircleSector(B_2, L, N) → s
  57. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  58. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  59. Segment(R', S) → f'''
  60. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  61. Vector(B_2, B') → u
  62. Translate(s, u) → s'
  63. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  64. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  65. Vector(C, C') → v
  66. Translate(s'', v) → s'''
  67. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  68. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  69. Vector(D, D') → w
  70. Translate(t, w) → t'
  71. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  72. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  73. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  74. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  75. Vector(B_2, B'_1) → v_1
  76. Vector(B_2, B'_3) → w_1
  77. Vector(B_2, B'_2) → i_1
  78. Circle(A, n - j_1) → h_2
  79. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  80. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  81. Vector(B_2, B'_4) → u_1
  82. Translate(s, u_1) → s'_1
  83. Line(B, f') → a_1
  84. Point(a_1) → U
  85. Angle(U, B, C) → α
  86. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  87. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  88. Line(A, B_2) → h_1
  89. Angle(B, A, D) → β
  90. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  91. Mirror(C, D) → C'_1
  92. Rotate(C, ((-π)) / 3, B_2) → C'_2
  93. Segment(B_2, C'_2) → f_1
  94. Segment(C, C'_2) → g_1
  95. Intersect(k, g_1, 1) → E
  96. Intersect(k, h) → T,
  97. CircleSector(C, T, E) → d_1
  98. Rotate(D, 15°, C) → D'_2
  99. Angle(D, C, D'_2) → γ
  100. Ray(C, D'_2) → k_1
  101. Rotate(C, (-(15°)), D) → C'_3
  102. Angle(C'_3, D, C) → δ
  103. Ray(D, C'_3) → m_1
  104. Intersect(k_1, m_1) → W
  105. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  106. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  107. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  108. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  109. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  110. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  111. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  112. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  113. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  114. Rotate(T, 15°, C) → T'
  115. Angle(T, C, T') → ε
  116. Ray(C, T') → q_1
  117. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  118. Vector(C, C'_8) → u_2
  119. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  120. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  121. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  122. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  123. Segment(A, A'_2) → j_2
  124. Segment(B_2, G_1) → l_2
  125. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  126. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  127. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  128. Polygon(H_1, I_1, C'_{10}) → poly6', c'_{10}, h_3, i_3
  129. Dilate(B_2, If[10 < e ≤ 11, 11 - e, 0], K_1) → C_1
  130. Segment(B_2, C_1) → n_1
  131. Rotate(B_2, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → D_1
  132. Rotate(C_1, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → E_1
  133. Segment(D_1, E_1) → p_1
  134. Rotate(D_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → F_1
  135. Rotate(E_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → N_1
  136. Segment(F_1, N_1) → r_1
  137. Rotate(F_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → O_1
  138. Rotate(N_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → P_1
  139. Segment(O_1, P_1) → s_1
  140. Polygon(B_2, K_1, L_1, M_1) → poly6, b_3, k_2, l_3, m_3
  141. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], K_1) → Q_1
  142. Vector(B_2, Q_1) → c_1
  143. Translate(s, c_1) → s'_2
  144. Segment(C, K_1) → e_1
  145. Line(A, M_1) → t_1
  146. Intersect(t_1, e_1) → R_1
  147. Segment(A, R_1) → f_2
  148. Dilate(C, If[15 < e ≤ 16, 16 - e, 0], K_1) → C'_6
  149. Dilate(A, If[16 < e ≤ 17, 17 - e, 0], M_1) → A'_3
  150. Dilate(M_1, If[17 < e ≤ 18, 18 - e, 0], R_1) → S_1
  151. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C_2) → T_1
  152. Vector(B_2, T_1) → v_2
  153. Translate(s, v_2) → s'_3
  154. Line(R_1, p_1) → g_3
  155. Point(g_3) → U_1
  156. Angle(U_1, R_1, K_1) → ζ
  157. Polygon(A, B_2, C, D) → poly7, a_4, b_4, c_4, d_4
  158. Dilate(K_1, If[15 < e ≤ 16, 16 - e, 0], M_1) → W_1
  159. Segment(K_1, W_1) → p_2
  160. Dilate(M_1, If[16 < e ≤ 17, 17 - e, 0], V_1) → Z_1
  161. Segment(M_1, Z_1) → e_2
  162. Line(A, K_1) → k_3
  163. Intersect(n_3, k_3) → A_2
  164. Dilate(V_1, If[18 < e ≤ 19, 19 - e, 0], C_2) → D_2
  165. Segment(V_1, D_2) → p_3
  166. Line(C_2, p_1) → q_3
  167. Point(q_3) → F_2
  168. Angle(F_2, C_2, V_1) → θ
  169. Rotate(s'_3, If[18 < e ≤ 19, ((e - 18) * θ), θ], T_1) → e_3
  170. Dilate(C_2, If[19 < e ≤ 20, 20 - e, 0], A_2) → E_2
  171. Segment(C_2, E_2) → r_3
  172. Dilate(A, If[20 < e ≤ 21, 21 - e, 0], V_1) → A'_5
  173. Segment(A, A'_5) → s_3
  174. Angle(A, V_1, A_2) → κ
  175. Rotate(A, If[21 < e ≤ 22, ((e - 21) * κ), κ], V_1) → A'_6
  176. Rotate(A'_5, If[21 < e ≤ 22, ((e - 21) * κ), κ], V_1) → G_2
  177. Segment(A'_6, G_2) → t_3

AI 绘图提示语参考

绘制一个动态演示的2正方形定理，包含两个正方形ABCD和A'B'C'D'，通过平移、旋转和缩放等变换，展示它们之间的关系。首先画出正方形ABCD，然后通过平移和旋转得到A'B'C'D'，接着构造一系列线段和圆，最后通过交点和中点构造出关键点，展示定理的证明过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。